Mathematics: analysis and approaches HL formula booklet —

Study edition

Mathematics:
analysis and
approaches

Higher Level formula booklet, annotated page for page. Practise with this copy and the official booklet will already feel familiar — same topics, same order, same page.

Level
Higher Level (HL)

First examination
2021

Topics
1 Number and algebra · 2 Functions · 3 Geometry and trigonometry · 4 Statistics and probability · 5 Calculus

How to use it
Every topic band, syllabus reference and page break matches the original, so the page you learn here is the page you turn to there.

Printing
A4, scale 100%, background graphics on. One sheet prints as one page.

Annotated study edition — always sit the exam with the official booklet your school issues.

Mathematics: analysis and approaches formula booklet Version 1.0 · re-creation

Mathematics: analysis and approaches HL formula booklet —

This page is intentionally blank.

Mathematics: analysis and approaches formula booklet blank

Mathematics: analysis and approaches HL formula booklet —

Index

T1 Number and algebra — HL 7 blocks 2 T2 Functions — HL 4 blocks 3 T3 Geometry and trigonometry — HL 9 blocks 4 T4 Statistics and probability — HL 9 blocks 7 T5 Calculus — HL 9 blocks 9

Reading the tables. The left column is the syllabus reference — the same number your teacher and the exam paper use. The middle column names the result; the right column is the formula itself, with its conditions underneath.

Prior learning. Topic 3 opens with the prior-learning results (areas, volumes, distance and midpoint) that the course assumes rather than re-teaches. They sit on page 4 because that is where the original puts them.

Mathematics: analysis and approaches formula booklet Contents

Mathematics: analysis and approaches Topic 1 · Number and algebra 2

Topic 1

Number and algebra — HL

1.2

The

n

th term of an arithmetic sequence

u_{n} = u_{1} + (n - 1) d

The sum of

n

terms of an arithmetic sequence

S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d)

S_{n} = \frac{n}{2} (u_{1} + u_{n})

1.3

The

n

th term of a geometric sequence

u_{n} = u_{1} r^{n - 1}

The sum of

n

terms of a finite geometric sequence

S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}, r \neq 1

1.8

The sum of an infinite geometric sequence

S_{\infty} = \frac{u_{1}}{1 - r}, | r | < 1

1.4

Compound interest

F V = P V \times {(1 + \frac{r}{100 k})}^{k n}

where

F V

is the future value,

P V

is the present value,

n

is the number of years,

k

is the number of compounding periods per year,

r

% is the nominal annual rate of interest.

1.5

Exponents and logarithms

a^{x} = b \Leftrightarrow x = \log_{a} b

where

a > 0

b > 0

a \neq 1

1.7

Exponents and logarithms

\log_{a} x y = \log_{a} x + \log_{a} y

\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y

\log_{a} x^{m} = m \log_{a} x

\log_{a} x = \frac{\log_{b} x}{\log_{b} a}

Exponential and logarithmic functions

a^{x} = e^{x \ln a}

\log_{a} a^{x} = x = a^{\log_{a} x}

where

a, x > 0

a \neq 1

1.9

Binomial theorem

n \in ℕ

(a + b)^{n} = a^{n} +^{n} C_{1} a^{n - 1} b + \dots +^{n} C_{r} a^{n - r} b^{r} + \dots + b^{n}

^{n} C_{r} = \frac{n!}{r! (n - r)!}

Mathematics: analysis and approaches formula booklet 2

Mathematics: analysis and approaches Topic 1 · Topic 2 3

1.10

Combinations

^{n} C_{r} = \frac{n!}{r! (n - r)!}

Permutations

^{n} P_{r} = \frac{n!}{(n - r)!}

Extension of binomial theorem,

n \in ℚ

(a + b)^{n} = a^{n} (1 + n (\frac{b}{a}) + \frac{n (n - 1)}{2!} {(\frac{b}{a})}^{2} + \dots)

1.12

Complex numbers

z = a + b i

1.13

Modulus–argument (polar) and exponential (Euler) form

z = r (\cos θ + i \sin θ) = r e^{i θ} = r c i s θ

1.14

De Moivre's theorem

{[r (\cos θ + i \sin θ)]}^{n} = r^{n} (\cos n θ + i \sin n θ) = r^{n} e^{i n θ} = r^{n} c i s n θ

Topic 2

Functions — HL

2.1

Equations of a straight line

y = m x + c; a x + b y + d = 0; y - y_{1} = m (x - x_{1})

Gradient formula

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

2.6

Axis of symmetry of the graph of a quadratic function

f (x) = a x^{2} + b x + c \Rightarrow axis of symmetry is x = - \frac{b}{2 a}

2.7

Solutions of a quadratic equation

a x^{2} + b x + c = 0 \Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}, a \neq 0

Discriminant

Δ = b^{2} - 4 a c

2.12

Sum and product of the roots of polynomial equations of the form

\sum_{r = 0}^{n} a_{r} x^{r} = 0

Sum is \frac{- a_{n - 1}}{a_{n}}; product is \frac{(- 1)^{n} a_{0}}{a_{n}}

Mathematics: analysis and approaches formula booklet 3

Mathematics: analysis and approaches Topic 3 · Geometry and trigonometry 4

Topic 3

Geometry and trigonometry — HL

Prior learning

Prior learning — HL

Area of a parallelogram

A = b h

where

b

is the base,

h

is the height

Area of a triangle

A = \frac{1}{2} (b h)

where

b

is the base,

h

is the height

Area of a trapezoid

A = \frac{1}{2} (a + b) h

where

a

and

b

are the parallel sides,

h

is the height

Area of a circle

A = π r^{2}

where

r

is the radius

Circumference of a circle

C = 2 π r

where

r

is the radius

Volume of a cuboid

V = l w h

where

l

is the length,

w

is the width,

h

is the height

Volume of a cylinder

V = π r^{2} h

where

r

is the radius,

h

is the height

Volume of a prism

V = A h

where

A

is the area of cross-section,

h

is the height

Area of the curved surface of a cylinder

A = 2 π r h

where

r

is the radius,

h

is the height

Distance between two points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}

Coordinates of the midpoint of a line segment with endpoints

(x_{1}, y_{1})

and

(x_{2}, y_{2})

(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

3.1

Distance between two points

(x_{1}, y_{1}, z_{1})

and

(x_{2}, y_{2}, z_{2})

d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}}

Coordinates of the midpoint of a line segment with endpoints

(x_{1}, y_{1}, z_{1})

and

(x_{2}, y_{2}, z_{2})

(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})

Volume of a right-pyramid

V = \frac{1}{3} A h

where

A

is the area of the base,

h

is the height

Mathematics: analysis and approaches formula booklet 4

Mathematics: analysis and approaches Topic 3 · Geometry and trigonometry 5

Volume of a right cone

V = \frac{1}{3} π r^{2} h

where

r

is the radius,

h

is the height

Area of the curved surface of a cone

A = π r l

where

r

is the radius,

l

is the slant height

Volume of a sphere

V = \frac{4}{3} π r^{3}

where

r

is the radius

Surface area of a sphere

A = 4 π r^{2}

where

r

is the radius

3.2

Sine rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Cosine rule

c^{2} = a^{2} + b^{2} - 2 a b \cos C

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}

Area of a triangle

A = \frac{1}{2} a b \sin C

3.4

Length of an arc

l = r θ

where

r

is the radius,

θ

is the angle measured in radians

Area of a sector

A = \frac{1}{2} r^{2} θ

where

r

is the radius,

θ

is the angle measured in radians

3.5

Identity for

t a n θ

\tan θ = \frac{\sin θ}{\cos θ}

3.6

Pythagorean identity

\cos^{2} θ + \sin^{2} θ = 1

Double angle identities

\sin 2 θ = 2 \sin θ \cos θ

\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ

3.9

Reciprocal trigonometric identities

\sec θ = \frac{1}{\cos θ}

c o s e c θ = \frac{1}{\sin θ}

Pythagorean identities

1 + \tan^{2} θ = \sec^{2} θ

1 + \cot^{2} θ = c^{o s e c 2} θ

Mathematics: analysis and approaches formula booklet 5

Mathematics: analysis and approaches Topic 3 · Geometry and trigonometry 6

3.10

Compound angle identities

\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B

\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B

\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Double angle identity for

t a n

\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}

3.12

Magnitude of a vector

| v | = \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}, where v = (\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix})

3.13

Scalar product

v \cdot w = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}, where v = (\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix}), w = (\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \end{matrix})

v \cdot w = | v | | w | \cos θ, where θ is the angle between v and w

Angle between two vectors

\cos θ = \frac{v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}}{| v | | w |}

3.14

Vector equation of a line

r = a + λ b

Parametric form of the equation of a line

x = x_{0} + λ l, y = y_{0} + λ m, z = z_{0} + λ n

Cartesian equations of a line

\frac{x - x_{0}}{l} = \frac{y - y_{0}}{m} = \frac{z - z_{0}}{n}

3.16

Vector product

v \times w = (\begin{matrix} v_{2} w_{3} - v_{3} w_{2} \\ v_{3} w_{1} - v_{1} w_{3} \\ v_{1} w_{2} - v_{2} w_{1} \end{matrix}), where v = (\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix}), w = (\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \end{matrix})

| v \times w | = | v | | w | \sin θ, where θ is the angle between v and w

Area of a parallelogram

A = | v \times w | where v and w form two adjacent sides of a parallelogram

3.17

Vector equation of a plane

r = a + λ b + μ c

Equation of a plane (using the normal vector)

r \cdot n = a \cdot n

Cartesian equation of a plane

a x + b y + c z = d

Mathematics: analysis and approaches formula booklet 6

Mathematics: analysis and approaches Topic 4 · Statistics and probability 7

Topic 4

Statistics and probability — HL

4.2

Interquartile range

I Q R = Q_{3} - Q_{1}

4.3

Mean,

\bar{x}

, of a set of data

\bar{x} = \frac{\sum_{i = 1}^{k} f_{i} x_{i}}{n}, where n = \sum_{i = 1}^{k} f_{i}

4.5

Probability of an event

A

P (A) = \frac{n (A)}{n (U)}

Complementary events

P (A) + P (A^{'}) = 1

4.6

Combined events

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Mutually exclusive events

P (A \cup B) = P (A) + P (B)

Conditional probability

P (A | B) = \frac{P (A \cap B)}{P (B)}

Independent events

P (A \cap B) = P (A) P (B)

4.7

Expected value of a discrete random variable

X

E (X) = \sum_{i = 1}^{k} x_{i} P (X = x_{i})

4.8

Binomial distribution

X \sim B (n, p)

Mean and variance of the distribution.

Mean

E (X) = n p

Variance

V a r (X) = n p (1 - p)

4.12

Standardized normal variable

z = \frac{x - μ}{σ}

4.13

Bayes' theorem

P (B | A) = \frac{P (B) P (A | B)}{P (B) P (A | B) + P (B^{'}) P (A | B^{'})}

P (B_{i} | A) = \frac{P (B_{i}) P (A | B_{i})}{P (B_{1}) P (A | B_{1}) + P (B_{2}) P (A | B_{2}) + P (B_{3}) P (A | B_{3})}

Mathematics: analysis and approaches formula booklet 7

Mathematics: analysis and approaches Topic 4 · Statistics and probability 8

4.14

Variance

σ ²

σ^{2} = \frac{\sum_{i = 1}^{k} f_{i} (x_{i} - μ)^{2}}{n} = \frac{\sum_{i = 1}^{k} f_{i} x_{i}^{2}}{n} - μ^{2}

Standard deviation

σ

σ = \sqrt{\frac{\sum_{i = 1}^{k} f_{i} (x_{i} - μ)^{2}}{n}}

Linear transformation of a single random variable

E (a X + b) = a E (X) + b

V a r (a X + b) = a^{2} V a r (X)

Expected value of a continuous random variable

X

E (X) = μ = \int_{- \infty}^{\infty} x f (x) d x

Variance

V a r (X) = E [(X - μ)^{2}] = E (X^{2}) - {[E (X)]}^{2}

Variance of a discrete random variable

X

V a r (X) = \sum (x - μ)^{2} P (X = x) = \sum x^{2} P (X = x) - μ^{2}

Variance of a continuous random variable

X

V a r (X) = \int_{- \infty}^{\infty} (x - μ)^{2} f (x) d x = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}

Mathematics: analysis and approaches formula booklet 8

Mathematics: analysis and approaches Topic 5 · Calculus 9

Topic 5

Calculus — HL

5.12

Derivative of

f (x)

from first principles

y = f (x) \Rightarrow \frac{d y}{d x} = f^{'} (x) = lim_{h \to 0} (\frac{f (x + h) - f (x)}{h})

5.3

Derivative of

x^{n}

f (x) = x^{n} \Rightarrow f^{'} (x) = n x^{n - 1}

5.6

Derivative of

s i n x

f (x) = \sin x \Rightarrow f^{'} (x) = \cos x

Derivative of

c o s x

f (x) = \cos x \Rightarrow f^{'} (x) = - \sin x

Derivative of

e^{x}

f (x) = e^{x} \Rightarrow f^{'} (x) = e^{x}

Derivative of

l n x

f (x) = \ln x \Rightarrow f^{'} (x) = \frac{1}{x}

Chain rule

y = g (u), where u = f (x) \Rightarrow \frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

Product rule

y = u v \Rightarrow \frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

Quotient rule

y = \frac{u}{v} \Rightarrow \frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

5.15

Standard derivatives
tan

x

f (x) = \tan x \Rightarrow f^{'} (x) = \sec^{2} x

sec

x

f (x) = \sec x \Rightarrow f^{'} (x) = \sec x \tan x

cosec

x

f (x) = c o s e c x \Rightarrow f^{'} (x) = - c o s e c x \cot x

cot

x

f (x) = \cot x \Rightarrow f^{'} (x) = - c^{o s e c 2} x

a^{x}

f (x) = a^{x} \Rightarrow f^{'} (x) = a^{x} (\ln a)

l o g_{a} x

f (x) = \log_{a} x \Rightarrow f^{'} (x) = \frac{1}{x \ln a}

a r c s i n x

f (x) = \arcsin x \Rightarrow f^{'} (x) = \frac{1}{\sqrt{1 - x^{2}}}

a r c c o s x

f (x) = \arccos x \Rightarrow f^{'} (x) = - \frac{1}{\sqrt{1 - x^{2}}}

a r c t a n x

f (x) = \arctan x \Rightarrow f^{'} (x) = \frac{1}{1 + x^{2}}

Mathematics: analysis and approaches formula booklet 9

Mathematics: analysis and approaches Topic 5 · Calculus 10

5.9

Acceleration

a = \frac{d v}{d t} = \frac{d^{2} s}{d t^{2}}

Distance travelled from

t_{1}

t_{2}

distance = \int_{t_{1}}^{t_{2}} | v (t) | d t

Displacement from

t_{1}

t_{2}

displacement = \int_{t_{1}}^{t_{2}} v (t) d t

5.5

Integral of

x^{n}

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1

Area between a curve

y = f (x)

and the

x

-axis, where

f (x) > 0

A = \int_{a}^{b} y d x

5.10

Standard integrals

\int \frac{1}{x} d x = \ln | x | + C

\int \sin x d x = - \cos x + C

\int \cos x d x = \sin x + C

\int e^{x} d x = e^{x} + C

5.15

Standard integrals, continued

\int a^{x} d x = \frac{1}{\ln a} a^{x} + C

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \arctan (\frac{x}{a}) + C

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \arcsin (\frac{x}{a}) + C, | x | < a

5.16

Integration by parts

\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x

or \int u d v = u v - \int v d u

Mathematics: analysis and approaches formula booklet 10

Mathematics: analysis and approaches Topic 5 · Calculus 11

5.11

Area of region enclosed by a curve and

x

-axis

A = \int_{a}^{b} | y | d x

5.17

Area of region enclosed by a curve and

y

-axis

A = \int_{c}^{d} | x | d y

Volume of revolution about the

x

y

-axes

V = \int_{a}^{b} π y^{2} d x

or V = \int_{c}^{d} π x^{2} d y

5.18

Euler's method

y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n}); x_{n + 1} = x_{n} + h

where

h

is a constant (step length)

Integrating factor for

y^{'} + P (x) y = Q (x)

e^{\int P (x) d x}

5.19

Maclaurin series

f (x) = f (0) + x f^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + \dots

Maclaurin series for special functions

e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots

\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots

\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots

\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots

\arctan x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots

Mathematics: analysis and approaches formula booklet 11

Mathematics:analysis andapproaches

Contents

Number and algebra — HL

Functions — HL

Geometry and trigonometry — HL

Prior learning — HL

Statistics and probability — HL

Calculus — HL

Mathematics:
analysis and
approaches